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Framed vs Unframed Two-dimensional languages

Framed vs Unframed Two-dimensional languages. Marcella Anselmo Natasha Jonoska Maria Madonia Univ. of Salerno Univ. South Florida Univ. of Catania ITALY USA ITALY. Two-dimensional (2dim) languages.

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Framed vs Unframed Two-dimensional languages

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  1. Framed vs Unframed Two-dimensional languages Marcella Anselmo Natasha Jonoska Maria Madonia Univ. of Salerno Univ. South Florida Univ. of Catania ITALY USA ITALY

  2. Two-dimensional(2dim)languages In the literature two kinds of 2dim languages • Sets of finite pictures Ex.L01= the set of finite pictures with one occurrence of symbol “1” at most and symbol “0” in the other positions • Tilings of the infinite plane Ex. Tiling of the infinite plane with one occurrence of symbol “1” at most and symbol “0” in the other positions Remark: The set of its finite blocks is L01

  3. Overview of the talk • Topic:Recognizable2dim languages • Motivation:In the literature • recognizable = (symbol-to-symbol) projection of local • with two different approaches • framed for finite pictures and • unframed for the infinite plan In this talk New “unframed” definition for “finite” pictures • Results of comparison framed vs unframed • with special focus on determinism and unambiguity Framed vs Unframed 2dim languages

  4. Local 2dim languages “Framed” approach • Generalization of local 1dim (string) languages • sharp () is needed to test locality conditions on the boundaries “Unframed” approach • Tiling of the (infinite) plane • No sharp is needed!

  5.          p = p =        • L islocalif there exists a finite set  of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of is in  (and we write L=L() )  p Local languages: LOC • finite alphabet,  **all pictures over , • L  ** 2dim language • To define local languages, identify the boundary of a picture p using a boundary symbol Framed vs Unframed 2dim languages

  6. 0 0 0 1 0 0 0  1 0 1 0 1  0   0 1 0 0  0 0 0    = 0    0         0 0 1  0 0  0 0 1  1 0  1 0     1  0  p = # # # # # 1 0 0 # 1 0 0 # 0 1 0 # 0 1 0 # p = 0 0 1 # 0 0 1 # # # # # # Example of local language Ld = the set of square pictures with symbol “1” in all main diagonal positions and symbol “0” in the other positions Framed vs Unframed 2dim languages

  7. Recognizable languages: REC • L is recognizable by tiling systemif L= (L’) where L’ is a local language and  is a mapping from the alphabet  of L’ to the alphabet of L • (, , , ) , where L’=L(), is called tiling system • REC is the family of two-dimensional languages recognizable by tiling system[Giammarresi, Restivo 91] Example: LSq = all squares over {a} is recognizable by tiling system. Set L’=Ld and (1)= (0)= a Framed vs Unframed 2dim languages

  8. Factorial local/recognizable languages • Factorial recognizable languages (FREC) are defined in terms of factorial local languages (FLOC) Do not care about the boundary of a picture! • L isfactorial localif there exists a finite set  of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of p is in  (and we write L=Lu() ) (throw away the … hat!!!) • L is factorial tiling recognizable if L= (L’) where L’ is a factorial local language and  is a mapping from the alphabet  of L’ to the alphabet of L (, ,, ) , where L’=Lu(), is called unborderedtiling system Framed vs Unframed 2dim languages

  9. e c c f e e f f e e c f f f a 1 1 b e e f f a a c f b b  = a 1 1 b a a b b g g g d d h e h h c g d d h g g h h g g g d d h h e h c Example of L in FREC L01 = the set of pictures with one occurrence of symbol “1” at most and symbol “0” in the other positions  Framed vs Unframed 2dim languages

  10. LOC and FLOC, REC and FREC   / / • L FLOC or L FREC implies L factor-closed • (i.e. L=F(L) where F(L) is the set of all factors of L) • L FLOC implies L LOC   (adding everywhere) • L FREC implies L REC • (as before) • FLOC LOC • Example: Ld LOC, not factor-closed • FREC REC • (as before) Framed vs Unframed 2dim languages

  11. Characterization of FLOC inside LOC and of FREC inside REC Proposition FLOC = LOC  Factor-closed Proof LLOC and L factor-closed implies L FLOC. Indeed  nofor F(L)=L (remove tiles with ) Proposition L  FREC iff L  REC and L=p(K) with K factor-closed local language Framed vs Unframed 2dim languages

  12. Determinism and unambiguity • “Computing”by a tiling system(, , , ) • Given a picture p** looking for p’ ** such that • (p’)=p (i.e. for a pre-image p’ of p) • Determinism • One possible next step • Unambiguity • One possible accepting computation Remark Usually Determinism implies Unambiguity Framed vs Unframed 2dim languages

  13. a b c d unique way to fill this position with a symbol of  whose projection matches symbol s s Determinism in REC: DREC Def. [A, Giammarresi, M 07] A tiling systemis tl-br-deterministicif  a,b,c   and s  ,  unique tile such that (d)=s. (Analogously tr-bl,bl-tr,br-tl -deterministictiling system) DREC languages that admit a tl-br or tr-bl or bl-tr or br-tl- deterministic tiling system Framed vs Unframed 2dim languages

  14.   / / / Unambiguity in REC: UREC Definition [Giammarresi,Restivo 92]A tiling system (, , , ) is unambiguous for L** if for any pL there is a unique p’  L’ such that (p’)=p (p’ pre-image of p). L ** is unambiguous if it admits an unambiguous tiling system. UREC= all unambiguous recognizable 2dim languages Proposition [A, Giammarresi, M 07] LOC DREC UREC REC Framed vs Unframed 2dim languages

  15. Ambiguity in REC Definition L ** is finitely-ambiguous if there exists a tiling system for L such that every picture pL has k pre-images at most (for some k >1). L is infinitely-ambiguous if it is not finitely ambiguous. Framed vs Unframed 2dim languages

  16. Determinism and unambiguity in FREC • DFREC = languages that admit a deterministic unbordered tiling system • UFREC = languages that admit an unambiguous unbordered tiling system • Finitely-ambiguous and infinitely ambiguousfactorial recognizable languages Framed vs Unframed 2dim languages

  17. Example Recall the example L01 -1 p = -1 The unbordered tiling system for L01 is deterministic but it is not unambiguous -1 Framed vs Unframed 2dim languages

  18. Example (continued) -1 p = Moreover it can be shown that L01 is an infinitely ambiguous factorial language. Framed vs Unframed 2dim languages

  19. Unambiguity in FREC Proposition.UFREC = FLOC Proof. If L  FLOC then  is the identity. If L  UFREC any symbol in has an unique pre-image and then  is a one-to-one mapping • Remarks. • UFREC is a very limited notion • DFREC does not imply UFREC A better suited definition of unambiguity is necessary Framed vs Unframed 2dim languages

  20. Frame-unambiguity (I) Definition An unbordered tiling system for L is frame-unambiguos at p  L if, once we fix a frame of local symbols in p, p has at most one pre-image. One pre-image at most p = Definition LFREC is frame-unambiguous if it admits a frame-unambiguous unbordered tiling system. Remark The frame of boundary symbols in UREC is replaced by a frame of local symbols Framed vs Unframed 2dim languages

  21. Frame unambiguity (II) -1 -1 In L01 p = L01 is frame-unambiguous Proposition L  DFREC implies L is frame-unambiguous Framed vs Unframed 2dim languages

  22. Ambiguity in REC vs ambiguity in FREC (I) Determinism Frame-Unambiguity Determinism Unambiguity In REC In FREC Determinism Unambiguity • There are languages • infinitely ambiguous • finitely-ambiguous • unambiguous • There are languages • infinitely ambiguous • unambiguous • (as far as we know) Framed vs Unframed 2dim languages

  23. Ambiguity in REC vs ambiguity in FREC (II) Remark Frame reduces the ambiguity degree • Finitely-ambiguous factorial in FREC and unambiguous in REC -1 Framed vs Unframed 2dim languages

  24. Ambiguity in REC vs ambiguity in FREC (III) Moreover • Infinitely factorial ambiguous in FREC and unambiguous in REC Framed vs Unframed 2dim languages

  25. Conclusions • Frame can enforce size and content of recognized pictures • Frame can reduce ambiguity degree Additional memory Factorial recognizable 2dim symbolic dynamical systems analogies and interpretations in symbolic dynamics Framed vs Unframed 2dim languages

  26. Grazie

  27. Conclusions Frame can enforce size and content of recognized pictures Frame can reduce ambiguity degree Additional memory Tilings of the plane 2dim symbolic dynamical systems analogies and interpretations in symbolic dynamics Note When sets of tilings are invariant under translations, in symbolic dynamics: Local Projection “shifts of finite type” “sofic shifts” Framed vs Unframed 2dim languages

  28. Decidability properties Proposition: It is decidable whether a given unbordered tiling system is unambiguous and whether it is deterministic. Proposition: It is undecidable whether a given unbordered tiling system is frame-unambiguous. Framed vs Unframed 2dim languages

  29. Removing tiles with # does not always work … • Given a tiling system for L  REC, this does not allow to recognize F(L) as element of FREC ExampleConsider Ldand the tiling system for it. Teta contains all the sub-tiles of T no  F(L) but • Given a tiling system for L=F(L)  REC, we cannot prove that this allow to recognize L as subset of FREC Framed vs Unframed 2dim languages

  30. Finite and infinite ambiguity in FREC Proposition: For any k >=1, there is a k-factorial-ambiguous language. Proposition: Unambiguous-FREC  (Col-UFREC Row-UFREC)  Finitely ambiguous FREC TOGLIERE? SI Proposition: (Col-UFREC  Row-UFREC)  DFREC  Frame-unambiguous FREC Framed vs Unframed 2dim languages

  31. Pictureor two-dimensional string over a finite alphabet: a b b c a c b a c b b a a b a • finite alphabet •  ** all 2dim rectangular words (pictures) over  • L **2dim language Two-dimensional Languages

  32. Local 2dim languages: first approach First approach (“framed” one) Generalization of local 1dim (string) languages 1dim: L= an1bm | n,m>0 is finite 2dim:

  33. Unambiguity in FREC (II) One pre-image UFREC Fix no local symbol Fix first column or first row of local symbols Fix two consecutive sides of local symbols DFREC Fix the frame New definition Framed vs Unframed 2dim languages

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